For completeness, we give a summary of the carré du champ method in integral form applied to the sphere with entropy methods and a nonlinear diffusion equation of fast diffusion or porous medium type. More details can be found in a series of earlier papers on manifolds (see [1]}, [2]}, [3]} or [4]}) with similar computations done in the perspective of uniqueness for positive solutions of elliptic equations (rigidity) that can be found in [5]}, [6]}. In the specific case of the sphere, it is possible to reduce the problem to a computation based on the ultraspherical operator (see [7]}, [8]}, [9]}, [10]}, [11]}). The method in the \(d=1\)  case is slightly different (see for instance [3]}) and results do not require the use of nonlinear diffusion equations as it is known for a long time from [13]}, [14]}, so this will not be discussed here. There are exceptional values of the exponent (\(p=6\)  if \(d=3\) , or \(p=3\)  if \(d=4\) ) which require a specific treatment that will not be detailed here, but the result can easily be recovered by considering the limit as \(p\rightarrow 6\)  if \(d=3\)  and \(p\rightarrow 3\)  if \(d=4\)  in the inequalities: see for instance [9]}. Here we adopt the presentation of [2]} and go along the lines of the proof of [4]}. We adopt simplified notations and write \(\rm {Id}\)  instead of using the standard metric, or simply write the Laplace-Beltrami operator as \(-\Delta \) . We split the method in two steps, with a first algebraic part closely related with the computations of [6]} and entropy methods applied to a parabolic equation in order to establish the inequalities. Because of the regularizing effects of the parabolic equation, there is no regularity issue in considering positive solutions and solutions will be considered as smooth without further notice. An approximation scheme has of course to be done for initial data in \(H^1({\mathbb {S}}^d,d\sigma )\) : see for instance [11]} for details in the case of the ultraspherical operator.
